Anant Singh set out to shoot a film about the life of Nelson Mandela. It took about a quarter-century—almost as long as South Africa's first black president spent in prison.

Mr. Singh broached the idea in the mid-1980s. The filmmaker was becoming more politically active and Mr. Mandela, the towering figure of the black-liberation movement, was well into his second decade in jail. In a message passed on by a mutual...